Junction of a Periodic Family of Elastic Rods with a 3d Plate. Part I

We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis $x_3$) and we investigate the limit behavior of this problem as the periodicity $\varepsilon$ and the radius r of the rods tend to zero. We use a decomposition of the displacement field in the rods of the form u=U+v where the principal part U is a field which is piecewise constant with respect to the variable $(x_1,x_2)$ (and then naturally extended on a fixed domain), while the perturbation v remains defined on the domain containing the rods. We derive estimates of U and v in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to $\varepsilon$ and r, of the constant in Korn’s inequality in a domain with such a rough boundary. To deal with the field v, we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate.